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Journal articles on the topic 'Chromatic Polynomials'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 February 2022

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1

Morgan, Kerri. "Galois groups of chromatic polynomials." LMS Journal of Computation and Mathematics 15 (September 1, 2012): 281–307. http://dx.doi.org/10.1112/s1461157012001052.

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AbstractThe chromatic polynomialP(G,λ) gives the number of ways a graphGcan be properly coloured in at mostλcolours. This polynomial has been extensively studied in both combinatorics and statistical physics, but there has been little work on its algebraic properties. This paper reports a systematic study of the Galois groups of chromatic polynomials. We give a summary of the Galois groups of all chromatic polynomials of strongly non-clique-separable graphs of order at most 10 and all chromatic polynomials of non-clique-separableθ-graphs of order at most 19. Most of these chromatic polynomials

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2

Wakelin, C. D. "Chromatic polynomials and ?-polynomials." Journal of Graph Theory 22, no. 4 (1996): 367–81. http://dx.doi.org/10.1002/(sici)1097-0118(199608)22:4<367::aid-jgt10>3.0.co;2-c.

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3

BIGGS, NORMAN. "CHROMATIC POLYNOMIALS FOR TWISTED BRACELETS." Bulletin of the London Mathematical Society 34, no. 2 (2002): 129–39. http://dx.doi.org/10.1112/s0024609301008931.

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This paper is concerned with the chromatic polynomials of ‘bracelets’: specifically, graphs constructed by taking n copies of a complete graph and linking them together in a ring. Using a sieve method, explicit formulae for the dominant and subdominant terms of the chromatic polynomial are obtained. Finally, a simple description of the effect of twisting the links is obtained.

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4

Robertson, Ian. "T-chromatic polynomials." Discrete Mathematics 135, no. 1-3 (1994): 279–86. http://dx.doi.org/10.1016/0012-365x(93)e0089-m.

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5

Ray, Nigel, and William Schmitt. "Ultimate chromatic polynomials." Discrete Mathematics 125, no. 1-3 (1994): 329–41. http://dx.doi.org/10.1016/0012-365x(94)90174-0.

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6

Rao, R. V. N. S., and J. V. Rao. "Classification of Algebraic Properties of Chromatic Polynomials." Journal of Scientific Research 5, no. 3 (2013): 469–77. http://dx.doi.org/10.3329/jsr.v5i3.11634.

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This manuscript attempts to introduce the concept of chromatic polynomials of total graphs using Mobius inversion theorem. In fact it studies various algebraic properties of chromatic polynomial using Mobius inversion theorem. Keywords: Bond lattice; Chromatic polynomial; Mobius function; Poset. © 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v5i3.11634 J. Sci. Res. 5 (3), 469-477 (2013)

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7

Farrell, E. J., and Earl Glen Whitehead. "Connections between the matching and chromatic polynomials." International Journal of Mathematics and Mathematical Sciences 15, no. 4 (1992): 757–66. http://dx.doi.org/10.1155/s016117129200098x.

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The main results established are (i) a connection between the matching and chromatic polynomials and (ii) a formula for the matching polynomial of a general complement of a subgraph of a graph. Some deductions on matching and chromatic equivalence and uniqueness are made.

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8

Levin, Alexander B. "On the set of Hilbert polynomials." Bulletin of the Australian Mathematical Society 64, no. 2 (2001): 291–305. http://dx.doi.org/10.1017/s0004972700039952.

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We characterise the set of all Hilbert polynomials of standard graded algebras over a field and give solutions of some open problems on Hilbert polynomials. In particular, we prove that a chromatic polynomial of a graph is a Hilbert polynomial of some standard graded algebra.

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9

Kishore, Anjaly, and M. S. Sunitha. "On injective chromatic polynomials of graphs." Discrete Mathematics, Algorithms and Applications 07, no. 03 (2015): 1550035. http://dx.doi.org/10.1142/s1793830915500354.

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The injective chromatic number χi(G) [G. Hahn, J. Kratochvil, J. Siran and D. Sotteau, On the injective chromatic number of graphs, Discrete Math. 256(1–2) (2002) 179–192] of a graph G is the minimum number of colors needed to color the vertices of G such that two vertices with a common neighbor are assigned distinct colors. The nature of the coefficients of injective chromatic polynomials of complete graphs, wheel graphs and cycles is studied. Injective chromatic polynomial on operations like union, join, product and corona of graphs is obtained.

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10

Borowiecki, Mieczysław, and Ewa Łazuka. "Chromatic polynomials of hypergraphs." Discussiones Mathematicae Graph Theory 20, no. 2 (2000): 293. http://dx.doi.org/10.7151/dmgt.1128.

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11

Bielak, Halina. "Roots of chromatic polynomials." Discrete Mathematics 231, no. 1-3 (2001): 97–102. http://dx.doi.org/10.1016/s0012-365x(00)00308-3.

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12

Drgas-Burchardt, Ewa, and Ewa Łazuka. "Chromatic polynomials of hypergraphs." Applied Mathematics Letters 20, no. 12 (2007): 1250–54. http://dx.doi.org/10.1016/j.aml.2007.02.006.

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13

Alikhani, Saeid, and Yee-Hock Peng. "Chromatic zeros and generalized Fibonacci numbers." Applicable Analysis and Discrete Mathematics 3, no. 2 (2009): 330–35. http://dx.doi.org/10.2298/aadm0902330a.

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In this article we consider the problem whether generalized Fibonacci constants can be zeros of chromatic polynomials. We prove that all 2n-anacci numbers and all their natural powers cannot be zeros of any chromatic polynomial. Also we investigate (2n + 1)-anacci numbers as chromatic zeros.

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14

Brenti, Francesco, Gordon F. Royle, and David G. Wagner. "Location of Zeros of Chromatic and Related Polynomials of Graphs." Canadian Journal of Mathematics 46, no. 1 (1994): 55–80. http://dx.doi.org/10.4153/cjm-1994-002-3.

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AbstractWe consider the location of zeros of four related classes of polynomials, one of which is the class of chromatic polynomials of graphs. All of these polynomials are generating functions of combinatorial interest. Extensive calculations indicate that these polynomials often have only real zeros, and we give a variety of theoretical results which begin to explain this phenomenon. In the course of the investigation we prove a number of interesting combinatorial identities and also give some new sufficient conditions for a polynomial to have only real zeros.

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15

Abebe Ashebo, Mamo, та V. N. Srinivasa Rao Repalle. "Fuzzy Chromatic Polynomial of Fuzzy Graphs with Crisp and Fuzzy Vertices Using α-Cuts". Advances in Fuzzy Systems 2019 (2 травня 2019): 1–11. http://dx.doi.org/10.1155/2019/5213020.

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Coloring of fuzzy graphs has many real life applications in combinatorial optimization problems like traffic light system, exam scheduling, register allocation, etc. In this paper, the concept of fuzzy chromatic polynomial of fuzzy graph is introduced and defined based on α-cuts of fuzzy graph. Two different types of fuzziness to fuzzy graph are considered in the paper. The first type was fuzzy graph with crisp vertex set and fuzzy edge set and the second type was fuzzy graph with fuzzy vertex set and fuzzy edge set. Depending on this, the fuzzy chromatic polynomials for some fuzzy graphs are

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16

Brown, Jason, and Aysel Erey. "New bounds for chromatic polynomials and chromatic roots." Discrete Mathematics 338, no. 11 (2015): 1938–46. http://dx.doi.org/10.1016/j.disc.2015.04.021.

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17

Zhang, Ruixue, and Fengming Dong. "Properties of chromatic polynomials of hypergraphs not held for chromatic polynomials of graphs." European Journal of Combinatorics 64 (August 2017): 138–51. http://dx.doi.org/10.1016/j.ejc.2017.04.006.

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18

Jackson, Bill. "A Zero-Free Interval for Chromatic Polynomials of Graphs." Combinatorics, Probability and Computing 2, no. 3 (1993): 325–36. http://dx.doi.org/10.1017/s0963548300000705.

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LetGbe a graph andP(G, t) be the chromatic polynomial ofG. It is known thatP(G, t) has no zeros in the intervals (−∞, 0) and (0, 1). We shall show thatP(G, t) has no zeros in (1, 32/27]. In addition, we shall construct graphs whose chromatic polynomials have zeros arbitrarily close to 32/27.

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19

Dong, Fengming. "New expressions for order polynomials and chromatic polynomials." Journal of Graph Theory 94, no. 1 (2019): 30–58. http://dx.doi.org/10.1002/jgt.22505.

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20

Allagan, Julian A., and David Slutzky. "Chromatic polynomials of mixed hypercycles." Discussiones Mathematicae Graph Theory 34, no. 3 (2014): 547. http://dx.doi.org/10.7151/dmgt.1750.

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21

Biggs, N. L., M. H. Klin, and P. Reinfeld. "Algebraic methods for chromatic polynomials." European Journal of Combinatorics 25, no. 2 (2004): 147–60. http://dx.doi.org/10.1016/s0195-6698(03)00095-7.

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22

Drgas-Burchardt, Ewa, and Ewa Łazuka. "On chromatic polynomials of hypergraphs." Electronic Notes in Discrete Mathematics 24 (July 2006): 105–10. http://dx.doi.org/10.1016/j.endm.2006.06.018.

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23

Alikhani, Saeid, and Mohammad A. Iranmanesh. "Chromatic Polynomials of Some Dendrimers." Journal of Computational and Theoretical Nanoscience 7, no. 11 (2010): 2314–16. http://dx.doi.org/10.1166/jctn.2010.1613.

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24

Lenart, Cristian, and Nigel Ray. "Chromatic polynomials of partition systems." Discrete Mathematics 167-168 (April 1997): 419–44. http://dx.doi.org/10.1016/s0012-365x(96)00245-2.

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25

Chia, G. L. "Some problems on chromatic polynomials." Discrete Mathematics 172, no. 1-3 (1997): 39–44. http://dx.doi.org/10.1016/s0012-365x(96)00266-x.

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26

Rodriguez, J., and A. Satyanarayana. "Chromatic polynomials with least coefficients." Discrete Mathematics 172, no. 1-3 (1997): 115–19. http://dx.doi.org/10.1016/s0012-365x(96)00274-9.

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27

Chia, G. L. "A bibliography on chromatic polynomials." Discrete Mathematics 172, no. 1-3 (1997): 175–91. http://dx.doi.org/10.1016/s0012-365x(97)90031-5.

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28

Haggard, Gary, and Thomas R. Mathies. "The computation of chromatic polynomials." Discrete Mathematics 199, no. 1-3 (1999): 227–31. http://dx.doi.org/10.1016/s0012-365x(98)00343-4.

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29

Van Bussel, Frank, Christoph Ehrlich, Denny Fliegner, Sebastian Stolzenberg, and Marc Timme. "Chromatic polynomials of random graphs." Journal of Physics A: Mathematical and Theoretical 43, no. 17 (2010): 175002. http://dx.doi.org/10.1088/1751-8113/43/17/175002.

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30

Ray, N., and W. Schmitt. "Coclosure operators and chromatic polynomials." Proceedings of the National Academy of Sciences 87, no. 12 (1990): 4685–87. http://dx.doi.org/10.1073/pnas.87.12.4685.

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31

Berceanu, Cornel. "CHROMATIC POLYNOMIALS AND K-TREES." Demonstratio Mathematica 34, no. 4 (2001): 743–48. http://dx.doi.org/10.1515/dema-2001-0402.

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32

Biggs, Norman. "Specht modules and chromatic polynomials." Journal of Combinatorial Theory, Series B 92, no. 2 (2004): 359–77. http://dx.doi.org/10.1016/j.jctb.2004.09.001.

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33

Simonelli, Italo. "Optimal graphs for chromatic polynomials." Discrete Mathematics 308, no. 11 (2008): 2228–39. http://dx.doi.org/10.1016/j.disc.2007.04.069.

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34

Eastwood, Michael, and Stephen Huggett. "Euler characteristics and chromatic polynomials." European Journal of Combinatorics 28, no. 6 (2007): 1553–60. http://dx.doi.org/10.1016/j.ejc.2006.09.005.

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35

Satyanarayana, A., and R. Tindell. "Chromatic polynomials and network reliability." Discrete Mathematics 67, no. 1 (1987): 57–79. http://dx.doi.org/10.1016/0012-365x(87)90166-x.

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36

Whitehead, Earl Glen. "Chromatic polynomials of generalized trees." Discrete Mathematics 72, no. 1-3 (1988): 391–93. http://dx.doi.org/10.1016/0012-365x(88)90231-2.

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37

Woodall, D. R. "An inequality for chromatic polynomials." Discrete Mathematics 101, no. 1-3 (1992): 327–31. http://dx.doi.org/10.1016/0012-365x(92)90613-k.

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38

Møller, Jesper M., and Gesche Nord. "Chromatic Polynomials of Simplicial Complexes." Graphs and Combinatorics 32, no. 2 (2015): 745–72. http://dx.doi.org/10.1007/s00373-015-1578-6.

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39

OXLEY, JAMES, and DOMINIC WELSH. "Chromatic, Flow and Reliability Polynomials: The Complexity of their Coefficients." Combinatorics, Probability and Computing 11, no. 4 (2002): 403–26. http://dx.doi.org/10.1017/s0963548302005175.

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We study the complexity of computing the coefficients of three classical polynomials, namely the chromatic, flow and reliability polynomials of a graph. Each of these is a specialization of the Tutte polynomial Σtijxiyj. It is shown that, unless NP = RP, many of the relevant coefficients do not even have good randomized approximation schemes. We consider the quasi-order induced by approximation reducibility and highlight the pivotal position of the coefficient t10 = t01, otherwise known as the beta invariant.Our nonapproximability results are obtained by showing that various decision problems

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40

ELLIS-MONAGHAN, J., and I. MOFFATT. "Evaluations of Topological Tutte Polynomials." Combinatorics, Probability and Computing 24, no. 3 (2014): 556–83. http://dx.doi.org/10.1017/s0963548314000571.

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We find new properties of the topological transition polynomial of embedded graphs, Q(G). We use these properties to explain the striking similarities between certain evaluations of Bollobás and Riordan's ribbon graph polynomial, R(G), and the topological Penrose polynomial, P(G). The general framework provided by Q(G) also leads to several other combinatorial interpretations these polynomials. In particular, we express P(G), R(G), and the Tutte polynomial, T(G), as sums of chromatic polynomials of graphs derived from G, show that these polynomials count k-valuations of medial graphs, show tha

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41

Alikhani, Saeid, and Roslan Hasni. "Algebraic Integers as Chromatic and Domination Roots." International Journal of Combinatorics 2012 (May 14, 2012): 1–8. http://dx.doi.org/10.1155/2012/780765.

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Let G be a simple graph of order n and λ∈ℕ. A mapping f:V(G)→{1,2,…,λ} is called a λ-colouring of G if f(u)≠f(v) whenever the vertices u and v are adjacent in G. The number of distinct λ-colourings of G, denoted by P(G,λ), is called the chromatic polynomial of G. The domination polynomial of G is the polynomial D(G,λ)=∑i=1nd(G,i)λi, where d(G,i) is the number of dominating sets of G of size i. Every root of P(G,λ) and D(G,λ) is called the chromatic root and the domination root of G, respectively. Since chromatic polynomial and domination polynomial are monic polynomial with integer coefficient

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42

Zhang, Ruixue, and Fengming Dong. "Problems on chromatic polynomials of hypergraphs." Electronic Journal of Graph Theory and Applications 8, no. 2 (2020): 241. http://dx.doi.org/10.5614/ejgta.2020.8.2.4.

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43

Gerling, Melanie. "Bivariate chromatic polynomials in computer algebra." Journal of Symbolic Computation 93 (July 2019): 183–99. http://dx.doi.org/10.1016/j.jsc.2018.06.006.

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44

Arif, Nabeel E., Roslan Hasni, and Saeid Alikhani. "Chromatic Polynomials of Certain Polyphenylene Dendrimers." Journal of Computational and Theoretical Nanoscience 9, no. 4 (2012): 560–63. http://dx.doi.org/10.1166/jctn.2012.2061.

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45

Brown, Jason I. "On the Roots of Chromatic Polynomials." Journal of Combinatorial Theory, Series B 72, no. 2 (1998): 251–56. http://dx.doi.org/10.1006/jctb.1997.1813.

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46

Biggs, Norman. "A Matrix Method for Chromatic Polynomials." Journal of Combinatorial Theory, Series B 82, no. 1 (2001): 19–29. http://dx.doi.org/10.1006/jctb.2000.2017.

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47

Baxter, R. J. "Chromatic polynomials of large triangular lattices." Journal of Physics A: Mathematical and General 20, no. 15 (1987): 5241–61. http://dx.doi.org/10.1088/0305-4470/20/15/037.

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48

Pitteloud, Philippe. "Chromatic Polynomials and the Symmetric Group." Graphs and Combinatorics 20, no. 1 (2004): 131–44. http://dx.doi.org/10.1007/s00373-003-0544-x.

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49

Buckingham, Paul. "p-Adic Roots of Chromatic Polynomials." Graphs and Combinatorics 36, no. 4 (2020): 1111–30. http://dx.doi.org/10.1007/s00373-020-02171-y.

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50

LUSE, KERRY, and YONGWU RONG. "A CATEGORIFICATION FOR THE PENROSE POLYNOMIAL." Journal of Knot Theory and Its Ramifications 20, no. 01 (2011): 141–57. http://dx.doi.org/10.1142/s021821651100867x.

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Given a graph, we construct homology groups whose Euler characteristic is the Penrose polynomial of the graph, evaluated at an integer. This work is motivated by Khovanov's work on the categorification of the Jones polynomial for knots, and the subsequent categorifications of the chromatic and Tutte polynomials for graphs.

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